Recovery – 2011 CSPG CSEG CWLS Convention 1

FX Cadzow / SSA Random Noise Filter: Frequency Extension

Alexander Falkovskiy*, Elvis Floreani, Gerry Schlosser

alex@absoluteimaging.ca

Absolute Imaging Inc., Calgary, AB, Canada

Summary

The application of FX Singular Spectrum Analysis (SSA) or Cadzow filtering approach for random noise

attenuation on seismic data started from the works of Ulrych et al (1988) on eigenimage filtering of seismic

data. Trickett furthered this work by applying eigenimage filtering to 3D data frequency slices and later

extended FX Cadzow filtering by forming a larger Hankel matrix of Hankel matrices in multiple spatial

dimensions (Trickett, 2003, 2009). We propose to add another dimension for creating extended Hankel

matrices. Instead of using a single frequency slice for composing the extended matrix, we propose to use a

range of frequency slices. This additional dimension of the matrix increases our statistics which improves

the filter quality. The synthetic examples illustrate the filter quality improvement compared to the

conventional FX Cadzow filter. Application of Frequency Extension (FE) filter in combination with FX

Cadzow filter on real data showed better noise reduction compared to only FX Cadzow filtering.

Introduction

FX Singular Spectrum Analysis (SSA) has been successfully applied to seismic data. For application to

seismic data, Trickett applied SSA separately on individual frequency slices. Later, following applications

outside seismic processing - Golyandina (Golyandina et al, 2001, 2007) and Dologlou (Dologlou et. al,

1996), Trickett furthered this method by extending this FX Cadzow filtering (Cadzow, 1988) by means of

forming a larger Hankel matrix of Hankel matrices in multiple spatial dimensions.

The purpose of this presentation is to introduce an additional dimension for composing the Hankel matrices.

Instead of using only spatial dimensions for composing these matrices we propose to add a frequency

dimension to create an extended matrix from a series of frequency slices. This Frequency Extension

approach improves filter quality through better statistics by utilizing these multi-frequency slices.

Theory

The following two examples show how an extended block matrix A may be created:

a. Extended matrix

A = where Ai =

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b. Extended matrix - an example

In Hybrid (C2-FX) filtering (Trickett, 2009) or 2D-Extension (Golyandina et.al, 2007) a block matrix A is

composed of sub-matrices (Ai) which may be constructed from neighboring shot gathers. Here each is a

complex Fourier coefficient for trace j from ensemble i. After matrix construction, rank reduction is

performed using singular value decomposition (SVD). Then Fourier coefficient for each trace is

recovered from the rank-reduced matrix by averaging over multiple entries of the same in A, and an

inverse Fourier transform returns the filtered traces.

Instead of composing these extended matrices only over space dimensions and for a single frequency slice,

we propose to create extended matrices from several frequency slices. To illustrate this FE approach, let us

use the same templates of the extended matrices as in examples above, but the meaning of subscripts will be

different. In the example of the extended matrix (a), the second subscript j in will denote the same trace

number as before, but the first subscript i will refer now to the sequential frequency slice number.

To show the validity of this FE filtering concept, let us consider a simple example in 2D FX domain. Given

a two dimensional event with constant dip, it will be represented in TX and FX domains as follows:

u(t,x) = Aδ(t-px) U(ω, x) = Ae-iωpx

(1)

where A is an arbitrary coefficient representing the amplitude of the event, t is time, x is a spatial coordinate,

ω is angular frequency and p is some coefficient representing the dip of the event. Let us assume that this

data is regularly sampled xk = kΔx and ωk =kΔω. Sacchi (Sacchi, 2009) presented an example showing that

the trajectory matrix built along the x coordinate in FX domain will have a rank of one, and, therefore, may

be approximated by a rank one matrix in rank reduction.

Similarly, we can build a trajectory matrix along the ω coordinate in FX domain and such trajectory matrix

will also have a rank of one.

Let the frequency series be

Fk = U(ωk, x) = Ae-i kΔω px

(2)

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For 7 samples, the trajectory matrix M is

= =

which illustrates that the frequency extension matrix M has a rank of one.

In the example above, assuming that we have only one linear event and random noise, FE filtering means

(1) find matrix M1 of rank r=1 nearest, in a least square sense, to the original trajectory matrix M composed

of a range of Fourier slices of the observed data

(2) recover FE-filtered traces by inverse Fourier transform from Fourier coefficients in M1.

M1 will have multiple entries of Fourier coefficients corresponding to the same trace, and an average value

of them will be used for the inverse Fourier transform in the same manner as in C2-FX filtering.

Similarly, it is possible to create FE filter with higher ranks and for extended block matrices.

Synthetic Examples

The synthetic examples (Fig. 1-5) show the comparisons between the standard Cadzow FX filter and FE

filtering approach. For FE filtering we used 2D extension in both spatial and frequency dimensions. The

frequency extension filter result looks cleaner than that after the Cadzow FX filter and its difference display

shows that more noise has been removed.

Figure 1: Input data.

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Figure 2: FX Cadzow filter, rank 2.

Figure 3: FE filter, rank 3.

Figure 4: Difference plot: Input – FX Cadzow. Figure 5: Difference plot: Input – FE filter.

The following synthetic examples (Fig. 6) show that FE filter produces better results when applied together

with the Cadzow FX filter. FE filter was used as a pre-filter for Cadzow FX to improve the final output. In

this case we applied FE filter with mild parameters (rank 5) to remove some noise but to make sure that it

would not remove the signal, and after that in was applied Cadzow FX with stronger parameters (rank 3).

Figure 6 shows that the result of such a combined application of the two filters FE + Cadzow FX looks

better than only Cadzow FX filtering.

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Figure 6: More synthetic examples.

Input data

Gadzow FX filter

Filter series: FE - Gadzow FX

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Real Data Examples

The Frequency Extension approach proposed in this paper was applied to 2D shot ensembles (data courtesy

of Olympic Seismic). Figure 7 shows one of the original shot records ordered by offsets, the results of

application on this record C2 -FX filter (ranks 2 and 3), FE filter (rank 4), FE combined with C

2 -FX filter,

and the difference plots. On these data, the results of Frequency Extension filtering were comparable to that

for C2 -FX filter.

However, when both filters are applied together, the shot record looks better – FE rank 4 followed by

C2

-FX rank 2. Like in synthetic data example, the basis for selection of parameters for these filters (ranks)

was to apply a mild filter first (FE with rank 4), and then apply a stronger second filter (C2

-FX with rank 2).

The results of stacking these data are shown in Figures 8-10: unfiltered structure stack, stack after

application of C2-FX filtering on shots and stack after application of FE combined with C

2–FX on shots.

The application of C2-FX filtering (Fig. 9) look much better then unfiltered stack (Fig. 8), as it was

expected, but combined filtering with FE and C2

-FX (Fig. 10) is cleaner than stack after only C2

-FX

filtering.

Conclusions

The results of using an additional frequency dimension in composing the extended matrices of the Cadzow

FX filters look promising. Application of proposed Frequency Extension filter together with C2 –FX filter

on the real 2-D data showed cleaner shot records and stack compared to only C2 –FX filtering. As in the

Hybrid Cadzow, this FE approach opens the way for composing larger extended matrices by utilizing more

dimensions which should result in a more accurate filter.

Acknowledgements

We would like to thank Absolute Imaging for giving us the resources and support to do this paper, Olympic

Seismic Ltd. and Murphy Oil Company Ltd. for permission to show their data.

References

Cadzow, J., 1988, Signal Enhancement – A Composite Property Mapping Algorithm: IEEE Transactions on Acoustics, Speech and Signal

Processing, 36, 49-62.

Canales, L.L., 1984, Random Noise Reduction: SEG Extended Abstracts, 525-527.

Dologlou, I., Pesquet, J.C., and Skowronski, J., 1996, Projection-based rank reduction algorithms for multichannel modeling and image

compression, Signal Processing, vol. 48, 97-109.

Golyandina, N., Nekrutkin, V., and Zhigljavsky, A., 2001, Analysis of Time Series Structure: SSA and Related Techniques: CRC Press

Golyandina, N., Usevich, K., and I.Florinsky, 2007, Filtering of Digital Terrain Models by two dimensional Singular Spectrum: International

Journal of Ecology & Development, Vol. 8, No. F07, 81-94.

Sacchi, M. D., 2009, FX Singular spectrum analysis: CSPG CSEG CWLS Convention, Abstracts, 392-395.

Trickett, S. R., 2003, F-xy Eigenimage Noise Suppression, Geophysics, 68, 751-759.

Trickett, S., 2009, Prestack Rank-Reduction-Based Noise Suppression: CSEG Recorder, November, 24-31.

Ulrych, T., Freire, S., and Siston, P., 1988, Eigenimage Processing of Seismic Sections, SEG, Extended Abstracts 7, 1261.

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Figure 7: Ensemble and Frequency Extension Cadzow filtering and difference plots.

Original

Difference

Filter: C2

-FX

rank 2

time

Filter: C2

-FX

rank 3

time

Filter: FE

rank 4

time

Filter: FE + C2

-FX

rank 4 FE

rank 2 C2

-FX

time

offset offset

Data courtesy of Olympic Seismic

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Figure 8: Unfiltered structure stack.

zoomed

Figure 9: Unmigrated stack: C2-FX rank 2 filter on shots.

zoomed

Figure 10: Unmigrated stack: FE rank 4 filter combined with C2–FX rank 2 filter on shots.

zoomed

Data courtesy of Olympic Seismic Ltd. and Murphy Oil Company Ltd.